WEBVTT

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Angle A is a circumscribed
angle on circle O.

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So this is angle
A right over here.

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Then when they say it's
a circumscribed angle,

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that means that the
two sides of the angle

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are tangent to the circle.

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So AC is tangent to
the circle at point

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C. AB is tangent to
the circle at point B.

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What is the measure of angle A?

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Now, I encourage you
to pause the video now

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and to try this out on your own.

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And I'll give you a hint.

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It will leverage the fact that
this is a circumscribed angle

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as you could imagine.

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So I'm assuming you've
given a go at it.

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So the other piece
of information

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they give us is that angle D,
which is an inscribed angle,

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is 48 degrees and it intercepts
the same arc-- so this

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is the arc that it intercepts,
arc CB I guess you could call

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it-- it intercepts this
arc right over here.

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It's the inscribed angle.

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The central angle that
intersects that same arc

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is going to be twice
the inscribed angle.

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So this is going
to be 96 degrees.

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I could put three markers here
just because we've already

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used the double marker.

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Notice, they both intercept
arc CB so some people would

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say the measure of
arc CB is 96 degrees,

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the central angle is 96
degrees, the inscribed angle

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is going to be half
of that, 48 degrees.

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So how does this help us?

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Well, a key clue is that angle
is a circumscribed angle.

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So that means AC and AB are
each tangent to the circle.

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Well, a line that is
tangent to the circle

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is going to be perpendicular to
the radius of the circle that

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intersects the circle
at the same point.

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So this right over here is
going to be a 90-degree angle,

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and this right over here is
going to be a 90-degree angle.

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OC is perpendicular to CA.

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OB, which is a radius,
is perpendicular to BA,

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which is a tangent line, and
they both intersect right

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over here at B. Now, this
might jump out at you.

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We have a quadrilateral
going on here.

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ABOC is a quadrilateral,
so its sides

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are going to add
up to 360 degrees.

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So we could know, we
could write it this way.

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We could write the
measure of angle A

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plus 90 degrees plus another
90 degrees plus 96 degrees

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is going to be equal
to 360 degrees.

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Or another way of thinking
about it, if we subtract 180

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from both sides, if we
subtract that from both sides,

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we get the measure of
angle A plus 96 degrees

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is going to be equal
to 180 degrees.

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Or another way of
thinking about it

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is the measure of angle A
or that angle A and angle O

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right over here-- you
could call it angle COB--

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that these are going to be
supplementary angles if they

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add up to 180 degrees.

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So if we subtract 96
degrees from both sides,

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we get the measure
of angle A is equal

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to-- I don't want to make that
look like a less than symbol,

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let make it-- measure of
angle-- this one actually

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looks more like a--
measure of angle A

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is equal to 180 minus 96.

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Let's see, 180 minus
90 would be 90,

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and then we subtract another
6 gets us to 84 degrees.